Sorry but your wrong, in no way was Gulf air ever associated with Dubai, Dubai started up emirates by leasing aircraft from PIA, and was started up to cause competition.
I believe you need to go a little further back in your history. Gulf Air was originally owned by Bahrain, Qatar, The UAE and Oman (I don't believe I left anyone out). They provided service not only between points in the Gulf but from the major cities in Gulf (including Dubai) to various international destinations. In the early to mid 80's, as part of its effort to grow and diversify its economy, Dubai asked for increased services but was refused. It was from this that the idea for a Dubai based airline started. You are correct in saying that the original flights were started with leased aircraft from PIA.